Aircraft: CRJ900
Phase of flight: Climb
Anomaly: Aircraft Equipment Problem Less Severe; Deviation / Discrepancy - Procedural Published Material / Policy

I was the Pilot Flying (PF) and my First Officer (FO) was the Pilot Monitoring (PM). We departed ZZZ. The weather was VMC throughout the entire flight. While climbing; still below our clearance of FL230 we received a Bleed Air Duct Warning. I told my FO that I had the radios and controls and for him to reference the QRH. While he opened the book to find the procedures I was told by ATC to climb to FL250 and to fly direct to the departure airport; so that's what we did. At this point I'm now back communicating with my FO and we observed that both bleed valves had closed. In this case the QRH required us to descent to 10;000 feet or the lowest safe altitude. We asked ATC do descend to 10;000 feet. ATC responded by clearing us to 24;000 feet. I told them that we needed 10;000 feet. ATC response was that he was working on it. We began our descent to 24;000 feet and moments later we received the clearance to 10;000 feet. During this time the FO and I closely monitored the cabin altitude as it was climbing; but pretty slowly. For this reason; we descended at a good rate but not so quickly as to make it uncomfortable. We continued through the QRH checklist with no problem and completed the procedures for unpressurized flight as well. Just before reaching our level off at 10;000 feet we received a Cabin Alt Caution message; and then as we leveled off we received a Cabin Alt Warning message. We never began the Immediate Action procedure for Cabin Alt Emergency Descent for the reason that the Cabin Pressurization messages popped up so close to 10;000 feet. Furthermore; the Cabin Altitude Warning message popped up multiple times after we leveled at 10;000. During the time of our descent we properly switched radios and controls back and forth from one another a number of times. We did this because I thought it appropriate that the passengers initially hear from the Captain. I tried to make the initial announcement to the passengers; however; the PA from the captain side was transmitting intermittently. For this reason; I requested the FO to make multiple announcement to inform the passengers of our status and any updates. I also had the FO inform the Flight Attendants of our situation and any updates. Furthermore; it was in the descent that we messaged company/dispatch [and] let them know what we were doing and ask if they wanted us to continue or to turn back for the departure airport. We had the fuel for either option. Finally; they notified us to return to base so we told ATC that [an alternate] would be our destination. The rest of the flight became very hot. By the time we got on the ground and then waited for a gate the temperature in the cabin had reached 36 degrees Celsius and 38 in the flight deck. Other than that; after the QRH was completed we had a safe return.Company was well aware of our situation and I wish that gates would be more readily available when an aircraft has to return to field. [Report narrative contained no additional information.] CRJ-900 flight crew reported a dual bleed valve failure during climbout and returned to departure airport.